## Abstract

It is often assumed that Boltzmann's kinetic equation (BKE) for the evolution of the velocity distribution function *f*(** r**,

**,**

*w**t*) in a gas is valid regardless of the magnitude of the Knudsen number defined by

*ϵ*≡

*τ*d ln

*ϕ*/d

*t*, where

*ϕ*is a macroscopic variable like the fluid velocity

**or temperature**

*v**T*, and

*τ*is the collision interval. Almost all accounts of transport theory based on BKE are limited to terms in

*O*(

*ϵ*)≪1, although there are treatments in which terms in

*O*(

*ϵ*

^{2}) are obtained, classic examples being due to Burnett and Grad. The

*mathematical*limitations that arise are discussed, for example, by Kreuzer and Cercignani. However, as we shall show, the

*physical*limitation to BKE is that it is

*not*valid for the terms of order higher than

*ϵ*because the assumption of ‘molecular chaos’, which is the basis of Boltzmann's collision integral, is an approximation that applies only up to first order in

*ϵ*.

Another difficulty with Boltzmann's collision integral is that it is defined at a point, so that the varying ambient conditions upon which transport depends must be found by Taylor series expansions along particle trajectories. This fails in a strong-field magnetoplasma where, in a single collision interval, the trajectories are almost infinitely repeating gyrations; we shall illustrate this by deriving a dominant *O*(*ϵ*^{2}) transport equation for a magnetoplasma that *cannot* be found from Boltzmann's equation.

A further problem that sometimes arises in BKE occurs when an external force is present, the equilibrium state being constrained by the stringent Maxwell–Boltzmann conditions. Unless this is removed by a transformation of coordinates, confusion between convection and diffusion is probable. A mathematical theory for transport in tokamaks, termed *neoclassical transport*, is shown to be invalid, one of several errors being the retention of an electric field component in the drift kinetic equation.

## 1. Introduction

Boltzmann's main aims were to ‘prove’ the Second law of Thermodynamics and to place Maxwell's equilibrium velocity distribution *f*_{0} on a sound basis, Maxwell's treatment being heuristic rather than rigorous (Chapman & Cowling 1970, p. 73), but he had no success with momentum and energy transport. A decade earlier, elementary transport theory based on Clausius' mean free path had been developed by Maxwell with some assistance from Clausius, who pointed out that Maxwell's first treatment of diffusive heat transport had included a small convective contribution. This error followed from not making the important distinction between the particle velocity ** w** measured in the laboratory frame and the (peculiar) velocity

**measured in a frame convected with the fluid velocity**

*c***, i.e.**

*v***=**

*w***+**

*v***. It was probably this error (still apparent today in some tokamak transport equations, see §6) that led Maxwell to develop his transfer equation using his velocity distribution function**

*c**f*(

**,**

*r***,**

*w**t*),where

*n*is the particle number density; is the average of

*ϕ*(

**,**

*r***,**

*w**t*) over

*f*; and the term on the r.h.s. is the change in

*ϕ*due to collisions. Let

*g*be the relative speed between colliding molecules, and if the repulsive force between molecules at a distance

*r*apart is proportional to

*r*

^{−ν}, in

*δϕ*/

*δt g*appears in the form

*g*

^{(ν−5)/(ν−1)}. Maxwell chose the algebraically convenient value

*ν*=5, believing that this was in agreement with observations.

Boltzmann's great contribution was to find an expression for the collision integral in the particle conservation relation,where is the phase-space convective derivative,and ** F** is the particle acceleration due to an external force.

Boltzmann's collision integral, , is based on binary collisions with the constraints that

the walls and other boundary discontinuities are sufficiently ‘distant’ (several mean free paths) to have negligible effects and

the velocity of a particle is uncorrelated, both with its position and with the initial velocity of any particle with which it is about to collide. Thus, the probability that a class ‘1’ molecule collides with a class ‘2’ molecule is assumed to be proportional to the product of their phase-space number densities,

*f*_{1}*f*_{2}. The assumption that the correlations between the colliding particles are negligible is termed the*hypothesis of molecular chaos*.

As we shall see, constraint (ii) prevents *O*(*ϵ*^{2}) accuracy being achieved in the transport equations. The assumption of *molecular chaos* is an approximation that holds only up to first order in *ϵ*, the simplest demonstration of which is provided by a stationary isothermal gas in a gravitational field (see §4).

There is a rather neglected way of checking the merit of a mathematical theory of transport, namely to judge the *physical* validity of limiting cases; those transport formulae that have unphysical limits must be dismissed. By ‘valid limits’ is meant physically realizable transport processes not specifically excluded in the derivation of their more general forms. The presence of unphysical limits can be traced back either to errors in the basic kinetic equation or to approximations made in obtaining the solutions, but deciding whether or not a formula is unphysical is not always straightforward. Clear examples of unphysical transport equations would be heat transport in the isothermal limit (in the absence of thermoelectric effects) and diffusive transport that depends on the choice of reference frame; we shall find in §5 that Burnett's *O*(*ϵ*^{2}) transport equations suffer from both of these defects.

Another problem arises in Boltzmann's kinetic equation (BKE) when an external force ** F**, acting equally on all particles, is present. Diffusivity

*D must*be calculated in a frame convected with the fluid, otherwise

**will contribute a frame-dependent component to**

*F**D*, which by definition should be independent of the choice of reference frame. Also, equilibrium in the presence of

**requires that the Maxwell–Boltzmann conditions (see Chapman & Cowling 1970, p. 76) be satisfied. These are(1.1)where**

*F**T*is the temperature; is the symmetrical part of ∇

**;**

*v**k*

_{B}is Boltzmann's constant;

*m*is the molecular mass; and

*n*

_{0}is the number density at

*ϕ*=0. These constraints require the fluid to be stationary to zeroth order in the Knudsen number power series expansion, i.e.

*v*_{0}=0 in

**=**

*v*

*v*_{0}+

*v*_{1}+

*v*_{2}+⋯. If it is not, the constraint must be removed, either by transforming the kinetic equation to the convected and accelerating frame, or by finding a reason for

**to be neglected.**

*F*For example, in the kinetic theory of an electron gas of density *ϱ*_{e} in a gravitational field, *F*_{e}=*ϱ*_{e}** g**−

*en*

_{e}(

**+**

*E*

*v*_{e}×

**), where**

*B***is the electric field and**

*E***is the magnetic induction. The equation of motion of the electron gas is(1.2)where**

*B*

*R*_{e}is the force density acting on the electron fluid due to collisions with the ions (ignoring viscosity) and

*p*

_{e}is the electron pressure. Thus, in steady conditions, we could remove

*F*_{e}provided that the zeroth order pressure gradient is negligible, that the conductivity is very large so that

*R*_{e}can be omitted, and that we are accelerating under gravity. But in any case, as we shall see in §4, pressure gradients are not compatible with Boltzmann's equation if terms in

*O*(

*ϵ*

^{2}) are of interest.

## 2. Boltzmann's collision integral

Boltzmann's formula for the collision integral reads as(2.1)where ** g**=

*w*_{1}−

*w*_{2}=

*g*

**is the relative speed and**

*k**α*(

**|**

*k***′;**

*k**g*) is the probability that the collision changes

**lying in d**

*k***into**

*k***′ lying in d**

*k***′. The term is proportional to the rate at which class ‘1’ particles are scattered out of a phase-space element d**

*k**ν*

_{1}due to direct collisions . The inverse collisions that scatter class ‘1’ particles

*into*d

*ν*

_{1}are obtained by simply reversing the direction of the collisions in , i.e. , and (2.1) follows from the assumption that the collisional encounter is locally

*reversible*, which yields(2.2)It is also assumed that no changes occur in the distribution functions over the penultimate collision intervals and also that the collision point is not accelerating.

While molecular chaos might be a good approximation before a collision, it is certainly not so after a collision, since the velocities and appearing in the product are merely the known velocities *w*_{1} and *w*_{2} reversed in direction. Given *f*_{1}(*w*_{1}), the distribution *f*_{2}(*w*_{2}) is assumed to be independent of *f*_{1} with all directions of the velocity *w*_{2} equally probable, but after the collision, with *f*_{1} and *f*_{2} known, the product is constrained by having a known argument , i.e. pure chaos has disappeared. This also means that should a particular velocity *w*_{2} be more probable than others, this feature is transferred to in the reverse direction without physical justification. We shall show in §4 that pure molecular chaos cannot exist when pressure gradients are present, i.e. there is always some correlation between *f*_{1} and *f*_{2}, and reversing velocities to obtain the arguments of and compounds the error.

With the help of his collision operator, Boltzmann was able to establish his famous *H*-theorem, which states that in a closed system ** S** defined in phase space, the functional(2.3)steadily decreases until it reaches a minimum value corresponding to an equilibrium state. He also showed that when

*H*is a minimum, , which is thus the necessary condition for equilibrium—a result known earlier to Maxwell. It follows that, subject to the conditions under which Boltzmann's collision integral is valid, Maxwell's distribution function is necessary and sufficient for equilibrium.

Chapman & Cowling (1970, p. 79) comment on molecular chaos as follows.Suppose that at a certain instant the velocity of every molecule in a mass of gas in a uniform state under no forces is reversed; the value of

*H* or is unaltered by this process. The molecules will now retrace their previous paths. Since, in general, ∂*H*/∂*t*<0 before the change, one infers that ∂*H*/∂*t*>0 after the change, which contradicts the *H*-theorem. Thus a paradox arises.The

*H*-theorem is, of course, a probability theorem. Its proof depends on probability concepts, e.g. in the definition of the velocity-distribution function and the calculation of the number of encounters of a given type. Thus the theorem is to be interpreted as implying, not that *H* for a given mass of gas must necessarily decrease during a given short interval, but that it is far more likely to decrease than to increase. Even so, the theorem appears inconsistent with reversibility, which implies that for every state of the gas for which *H* is decreasing there is one for which *H* is increasing equally rapidly.The following considerations help towards resolving the paradox. The argument leading to the expression for (d

*f*/d*t*)_{col} assumes molecular chaos before encounters, i.e. that the velocities of two molecules just before they collide are uncorrelated. This is reasonable because in gases at moderate densities the two molecules will have come from regions far enough apart to rule out the possibility of their having recently influenced each other. The velocities of the molecules immediately after encounter are not, however, in the same sense uncorrelated. If the velocities of all the molecules are suddenly reversed, encounters are described in the reverse sense, and one is no longer considering a state with molecular chaos before encounters. Thus the value found earlier for ∂*H*/∂*t* no longer applies. This in part resolves the difficulty; on physical grounds, states characterized by molecular chaos before collisions are far more probable than states deviating widely from such chaos. But the mathematical difficulty remains that to each state characterized by molecular chaos there corresponds one (that with velocities reversed), not so characterized.

Boltzmann's equation can be put into a form resembling the Bhatnagar–Gross–Krook (BGK; Bhatnagar *et al*. 1954) kinetic equation,whereandIf is approximated by *f*_{0}, we have the very useful BGK kinetic equation. This approximation can be compensated almost completely by adopting two distinct collision intervals: *τ*_{1} for momentum transport and *τ*_{2}=1.5*τ*_{1} for energy transport, a choice that yields the correct Prandtl number (*Pr*≡*c*_{p}*μ*/*κ*, where *c*_{p} is the specific heat at constant pressure and *κ* is the thermal conductivity). Constraints on BGK also apply to BKE, and this is also true of the Fokker–Planck approximation to BKE.

## 3. The Chapman–Enskog solution

It is not possible to solve the nonlinear, integro-differential equationexactly. A brief outline of the Chapman–Enskog iterative solution is as follows. The Knudsen number, , is adopted as the expansion parameter. To zeroth order in *ϵ*,where *f*_{0} is the Maxwellian distribution.

Not only *f*, but also must be expanded in a series in *ϵ*. Hilbert's similar approach1 failed because he did not expand the time derivative, but it is possible that the Chapman–Enskog expansion goes too far with the time-derivative expansion (see Cercignani 1990, p. 119). We havewhich are substituted into the kinetic equation; terms of the same order are equated, yielding the sequence of linear equationswhich are solved in turn for *f*_{1}, *f*_{2}, ….

In a neutral gas, the distribution function is found to be (for example see Woods 1993)(3.1)where *τ*_{2} and *τ*_{1} are the collision intervals whose values remain to be determined, the superscript circle denotes the deviator (no trace) and *ν*=*c*/*C*, . Provided the momentum flux is *collided* flux, the pressure tensor is(3.2)where denotes the unit tensor. Otherwise is the momentum flux, which by Newton's Second Law is zero. If *t*^{*} is the time it takes an average molecule to reach the boundary, the wall pressure is . Note that if the gas is collisionless, *p* does *not* equal *nk*_{B}*T*.

The heat flux vector iswhere *κ*=(5/2)*Rpτ*_{2}=(15/4*R*)*pτ*_{1}=(5/2)*c*_{v}*μ*, since . This also must be *collided* flux, otherwise *T* will not have the properties required of a scalar temperature function.

## 4. The effect of pressure gradients

Maxwell's transfer equation, without the acceleration term, isLet *ϕ*=*m*** c**, so that , and by the assumed conservation of momentum (and energy) during binary collisions,

*δϕ*/

*δt*=0. This moment therefore tells us nothing about , since Boltzmann's collision integral vanishes whether or not there are collisions. To identify as a pressure tensor and not merely as an unchanging momentum flux, it is necessary to add some physics, namely that there are sufficient collisions in the region under consideration. Boltzmann's approximation, i.e. no collective effects and no correlations between colliding particles, is successful to first order in

*ϵ only*because it occurs in the collision integral, which is the term of lowest order in

*ϵ*.

Consider the case of an isothermal, stationary gas in a gravitational field, as illustrated in figure 1. Equilibrium is maintained by particle collisions transmitting a force upwards against gravity, yet because the Boltzmann collision integral is merely number density conservation, the distribution functions of the colliding particles are Maxwellian and being isotropic cannot supply the required upward pressure force. There has to be a greater probability that colliding particles are from the region below the actual collision point, as shown in figure 1. The conclusion is that *molecular chaos does not exist in the presence of an external force*, a point not quite recognized by Chapman & Cowling (1970, p. 80). Their resolution of what they term a paradox is as follows.The reversibility paradox can be related to another occurring in the kinetic theory. Consider, for example, the following paradox relating to an atmosphere in a steady state under gravity. Any one molecule has a constant downward acceleration due to gravity, and since with the [Maxwellian velocity distribution] velocities in all directions are equally likely, collisions with other molecules may impede, but will not completely destroy, the downward motion of the molecule. Hence the gas as a whole must descend; that is, it cannot be in a steady state.

This […] paradox is easily resolved. If the atmosphere is held up against gravity, it must be held up by a surface, and collisions with this surface interrupt the steady descent. To see how the steady state is maintained at a level well above this surface, consider two neighbouring horizontal planesAandB, of whichAis aboveB. Because of the action of gravity it is more probable that a molecule which at the beginning of a short interval is atAwill sink toBduring that interval, than that a molecule initially atBwill rise toA. As, however, the density of molecules is greater atBthan atA, the smaller proportion of the molecules initially atBwhich rise toAcan exactly balance the larger proportion of molecules initially atAwhich sink toB. Hence each molecule can tend to diffuse downward with a certain velocity, and yet the mean velocity of molecules at a given point can vanish.

But there are *only* collisions with other particles to balance gravity, and there is no doubt that a steady state can be found with constant temperature and stationary fluid. So this ‘resolution’ artificially separates the downward diffusion due to gravity from the isotropic diffusion due to the number density gradient. The simple equilibrium condition under a gravitational force, , does *not* follow from Boltzmann's equation, but from a separate assertion concerning the existence of collisions that on average are directed more upwards than downwards. Our conclusion is that, at least in the presence of a pressure gradient, BKE is not valid beyond *O*(*ϵ*)—although this conclusion is contended by Alexeev (2004), among others.

## 5. The Burnett equations

Another way of checking BKE is to examine the *O*(*ϵ*^{2}) terms derived from it to see that they make physical sense. The second-order terms in ** q** and

*π*, attributed to Burnett, arewith ,

*θ*

_{2}=45/8,

*θ*

_{3}=−3,

*θ*

_{4}=3,

*θ*

_{5}=(35/4)+

*s*andwith , , , , , , where

*s*is defined by the viscosity relation

*μ*∝

*T*

^{s}.

The seven terms in *q*_{2} and *π*_{2} labelled (i)–(vii) owe their presence to the neglect of fluid frame accelerations in defining diffusion. Since , where , the time derivative appearing in *q*_{2} is frame *dependent*. But, diffusion must be independent of the choice of reference frame, and for frame indifference the correct combination for term (i) is . Similar remarks apply to the term (iv) in *π*_{2}. Terms (ii) and (iii) imply that there can be a heat flux in isothermal conditions, which in the absence of thermoelectric effects, is impossible.

Turning to *π*_{2}, terms (v)–(vii) imply the existence of viscous forces in the absence of fluid shear, which is unphysical. It follows that of the 11 second-order terms, 7 are spurious and arise because Boltzmann's collision integral is based on molecular chaos. For the heat flux vector we are left withbut even here error remains as the correct values of *θ*_{i} (*i*=1, 2, 5) are not those in Burnett's formula, e.g. on physical grounds we would expect the second-order thermal conductivity to be , not , which is far too large to make physical sense.

The derivation of the Burnett equations is at least *mathematically* sound, leaving the following possibilities: Boltzmann's collision integral is accurate only up to terms in *O*(*ϵ*), the Chapman–Enskog expansion procedure is at fault as argued by Cercignani (1990, p. 119), or both faults are present. Cercignani writes:Thus we may expect the Chapman–Enskog theory to be much more accurate than the Hilbert theory; on the other hand, if we consider higher-order approximations of the Chapman–Enskog method, we obtain differential equations of higher order (the so-called Burnett and super-Burnett equations), about which nothing is known, not even the proper boundary conditions. These higher-order equations have never achieved any noticeable success in describing departures from continuum fluid mechanics; furthermore, a preliminary treatment of the connection problem for boundary layers seems to yield the same number of boundary conditions at any order of approximation …, while the order of differentiation increases. These and other facts seem to suggest that the Chapman–Enskog theory goes too far in the direction of taking into account the contributions of different orders in

*ϵ* to the time derivative …the practical result is that we complicate the equations by inserting details which are either irrelevant or nonexistent. The fact that the Chapman–Enskog expansion can bring in solutions which are simply non-existent is not strange …

We conclude that BKE is not accurate when terms in *O*(*ϵ*^{2}) are significant, e.g. for *ϵ* larger than approximately one-third, the value achieved in strong shock waves. It follows that when terms in *O*(*ϵ*^{2}) are considered to be significant, we must either try to modify Boltzmann's collision integral as in Woods (1993) or revert to mean free-path methods carefully extended to include *O*(*ϵ*^{2}) terms (see §9).

Transforming the BGK kinetic equation, namelyto the convected frame and allowing for the change in velocity due to the pressure impulse ** P**=−∇./

*ϱ*, and the change in during the mean free-path transit of the molecules being scattered into the phase-space element, we getwhere

*D*is the time derivative in the convected frame (see Woods 1993, p. 112). The acceleration is given by , and is due to the origin from which

**is defined changing continuously by fluid shear. This kinetic equation removes the unphysical terms from the Burnett equations, but still gives an excessively large value for**

*c**θ*

_{5}.

An equivalent modification is required to ‘correct’ Boltzmann's equation, at least to give *O*(*ϵ*^{2}) accuracy. But to achieve this is much too algebraically complex and in any case unlikely to have much value. The situation is rather different in plasma theory with strong magnetic fields, in which case the Knudsen number expansion applied to Boltzmann's equation yields a series for the heat flux orthogonal to the magnetic field of the type , where , in which *ω*_{c} is the cyclotron frequency and *τ* is the collision interval. In tokamaks *ϖ*∼10^{7} for the electron gas, whereas *ϵ*∼10^{−2}. As we shall show in §8, this expansion fails to find the second-order term of the form *ϵ*^{2}/*ϖ*, which is orders of magnitude larger than the first-order term (*ϵ*/*ϖ*^{2}), and which explains why tokamaks lose their energy so rapidly (Woods 2006).

## 6. Neoclassical transport theory

There is a theory for laminar (i.e. non-turbulent) transport in tokamaks known as *neoclassical theory*, which despite its failure to explain losses from tokamaks is still frequently referenced in the literature. It is obtained as follows. The kinetic equation for the ions or electrons in a magnetoplasma is(6.1)In strong magnetic fields ** B**, particles of mass

*m*and charge

*ξ*gyrate at a frequency

*ω*

_{c}=

*ξB*/

*m*in small circles about centres called

*guiding centres*. Letting , then the average radius of these circles is . The average of (6.1) taken around these circles is called the drift kinetic equation. The averaging is subject towith the result,2(6.2)where is the velocity distribution of the guiding centres and

*v*_{g}is their average velocity.

There are two serious errors; firstly, the retention of a force (*E*_{∥}) that acts on the particles collectively means that the distinction between diffusion and convection is lost and secondly, the gyro-average of ** w**×

**.∂**

*B**f*/∂

**is not in fact zero, but is proportional to (**

*w**ω*

_{c}

*τ*)

^{−1}, which for electrons is approximately 10

^{7}times larger than the term retained.

To compound these mistakes in the development of the theory, there is the strange notion that pressure gradients can act on particles, but *not on their associated guiding centres*. See Hazeltine & Meiss (1992, p. 145), where this curious statement can be found as follows.More generally, however, the distinction between guiding-centre motion and plasma motion is crucial. It explains, for example, the diamagnetic current of (3.63)3: no guiding centre executes a grad-P drift! Thus (152) , which we will call the ‘magnetization law’, has pervasive importance.

Surprisingly the magnetization law is not always appreciated. One can find in the literature confusions between

*v*_{gc} and ** v**, as well as misunderstandings concerning

**(the magnetization)4.**

*M*Of course, what happens to particles must also happen to the guiding centres, which by definition *shadow* them closely, never more than a distance *r*_{L} away; in tokamaks *r*_{L} is only millimetres for electrons. The fact that guiding centres cannot collide with each other is irrelevant.

Since in the *convected* reference frame and ignoring fluid shear, (6.2) becomesand *E*_{∥} has no role in the *diffusion* of energy and mass in tokamaks. The neoclassical transport equations derived by Rosenbluth *et al*. (1972) are based on (6.2); the resulting formulae are a mixture of diffusion and convection:(6.3)(6.4)andwhere *n* is the particle number density; *τ*_{e} and *τ*_{i} are collision intervals; *q* is the safety factor; *B*_{θ} is the poloidal magnetic field; *Γ* is the particle flux in the radial direction; and are the electron and ion radial heat fluxes; and *ϵ* is the ratio of the distance *r* from the minor axis of the torus to its major radius, *R*.

The terms involving *E*_{∥} do not appear in valid transport equations—there are no such terms in the scholarly texts of Chapman & Cowling (1970) and Ferziger & Kaper (1972), and even with *E*_{∥} removed, the neoclassical equations are evidently invalid. Valid transport equations must remain so for limiting cases of the driving thermodynamic forces. Consider isothermal conditions with *n*′<0 and *E*_{∥}=0. By (6.3) the particles will diffuse *outwards*, whereas by (6.4) the thermal energy carried by these particles will be transported *inwards*, an unphysical mismatch, which means that (6.3) and (6.4) are wrong. A similar conclusion follows if we retain the convection term containing *E*_{∥} and set *n*′ equal to zero—an inward mass flux is wrongly associated with an outwards energy flux. In fact, convection in tokamaks is outwards, both in theory and by observation.

In addition to the above fluxes, neoclassical theory predicts a parallel electric current,(6.5)where(6.6)is known as the *bootstrap current*; several approximate expressions are available for the factor *g*, which is not unity owing to particle trapping in banana orbits. There are several arguments that show that *j*_{b}=0 (Woods 2006, pp. 74–76); perhaps the most obvious is to consider the limiting case *E*_{∥}=0, which is not excluded at any point in the theory. Indeed, it is argued by Helander & Sigmar (2002) that the bootstrap current is perhaps the most important result of neoclassical theory. They claim that it has the merit of arising spontaneously in response to radial gradients and since it is in the same direction as the ohmic current, it helps to confine the plasma by the additional poloidal field that it generates. They speculate that the improved confinement may cause the pressure gradient to steepen further, leading to still larger values of *j*_{b} and that if this process continues, this current could even constitute most of the total plasma current.

Apart from *direction*, *j*_{b} is independent of the electric field. Hence, if *E*_{∥}→0, so that *j*_{∥}→*j*_{b}, unless *j*_{b} is zero, the bootstrap current has an unphysical singularity in direction, i.e. while its magnitude is assumed to be finite, its direction is arbitrary, which of course must mean that it is zero.

## 7. Elementary theory of second-order transport

For a neutral gas, the elementary theory is based on the transport of macroscopic properties such as fluid temperature, *T*, and momentum flux, *ϱ*** v**, over a distance of a mean free path

**=**

*λ**τ*

**, which may differ slightly in magnitude depending on the property being transported. A particle**

*c**p*moving between a point

*Q*

_{1}(

**−**

*r**τ*

**,**

*c**t*−

*τ*) to another point

*Q*

_{2}(

**,**

*r**t*) transports properties acquired at

*Q*

_{1}to

*Q*

_{2}, i.e.

*p*is ‘labelled’ with the temperature and fluid momentum at

*Q*

_{1}and transports these properties to

*Q*

_{2}, where a collision completes the process. In a plasma, where grazing collisions are dominant, this is a continuous process, but the mechanism is not different in principle, so we can retain the neutral gas terminology. Thus, the change in temperature is(7.1)

To extend elementary kinetic theory to a magnetoplasma, we need to allow for the fact that the straight trajectories between collisions in a neutral gas are replaced by circular orbits about the field lines. By strong magnetic fields, we mean fields for which there are a great number of gyrations per collision interval, i.e. the parameter *ϖ*≡*ω*_{c}*τ* is very much larger than unity. In this case, the projections of particle orbits on to a plane normal to ** b** are tight, repeating circles; therefore, the simple notion of a physical source, lying a mean free path back along the trajectory, that labels the particle with its temperature and fluid velocity, needs to be revised. It is evidently necessary to relate particle speeds at

*Q*(

**,**

*r**t*) to macroscopic conditions at some point

*P*lying

*within*the circular path.

Consider the particle *p*, shown in figure 2, that has just passed through *Q*(** r**,

*t*) with the peculiar velocity , where is a unit vector. From the equation of motion, , its guiding centre

*G*(

**,**

*X**t*) is at(7.2)Typically

*p*may gyrate about

*G*millions of times per collision interval. The distribution function from which its peculiar velocity is chosen, i.e. the one that labels

*p*, has macroscopic variables obtained by averaging over the orbits. When

*ϖ*is very large, the ambient temperatures at all points on the orbits have equal weight in this averaging process, which means that it will yield the temperature at the guiding centre itself, subject to the usual collision interval time delay. Hence, in this case, the velocity distribution from which

*p*'s

*present*peculiar velocity is chosen is based on the temperature and fluid velocity that

*G had*one collision interval earlier.

Letting ** u** and

**be the velocities of**

*v**G*and

*Q*in the laboratory frame, then since at time

*t*,

*Q*is a vector distance

**from**

*a**G*, at the time

*t*−

*τ*this distance was

**−**

*a**τ*(

**−**

*v***). Hence, if the guiding centre was at and**

*u*

*v*_{g}denotes the fluid velocity at

*G*(

**,**

*X***), then(7.3)Now**

*t***=**

*v*

*v*_{g}+

**.∇**

*a***+**

*v**O*(

*a*

^{2}), whence expanding ∇

**we get(7.4)where**

*v***=(1/2)∇×**

*Ω***is the fluid spin and is the deviator (traceless symmetrical part of ∇**

*v***).**

*v*The frame-indifferent heat flux ** q** is defined in a frame rotating with the fluid element, in which frame

**=0, and therefore**

*Ω***cannot contribute to**

*Ω***. The Knudsen number is , so the term**

*q**τ*|

**∇.**

*a***| is**

*v**O*(

*ϵ*) smaller than |

**| in (7.3) and being parallel to it, can be omitted. In the absence of gradients in**

*a***,**

*B***=**

*u***+**

*v*

*c*_{∥}and therefore the effective value of

*X*_{S}is , which can be written as(7.5)this gives a good approximation to the location of the labelling source

*G*

_{S}.

The arguments yielding (7.1) can now be applied to the magnetoplasma, except that instead of a source *Q* at ** r**−

*τ*

**, we now have a source**

*c**G*

_{S}at

*X*_{S}. By (7.5), this is equivalent to the replacement . Therefore,

*τ*

**.∇**

*c**T*in (7.1) is replaced bywhich amounts to substitutingfor ∇

*T*in Fourier's Law,

**=−**

*q**κ*∇

*T*. It follows that in very strong magnetic fields the heat flux vector is given by

From standard kinetic theory *κ*=5*kpτ*/2*m* and *κϖ*^{−1}=5*k*_{B}/2*ξB*, whence(7.6)In order, the terms on the r.h.s. in this equation are the ‘parallel’, ‘transverse’ and ‘second-order’ heat fluxes. Their coefficients have magnitudes of *O*(*ϵ*), *O*(*ϵ*/*ϖ*) and *O*(*ϵ*^{2}/*ϖ*), and with typical laboratory values of *ϵ*∼0.1, *ϖ*∼10^{6}, they have relative magnitudes of 1, 10^{−6} and 10^{−8}. The parallel heat flux is dominant, which means that in a tokamak the temperature is almost constant on given magnetic fields lines.

The classical theory for cross-field transport gives (see (8.8) below)(7.7)which from the values given above is *O*(*ϵ*/*ω*^{2})∼10^{−13}, i.e. the new term in (7.6) is orders of magnitude larger than the classical value.5

## 8. Failure of Boltzmann's equation in tokamak transport

The equation of motion for a particle in a convected frame is (Woods 2004, p. 177)(8.1)and therefore the kinetic equation in the convected frame is(8.2)where(8.3)and *p*=*nk*_{B}*T*. Thus,(8.4)

Letting *φ*≡(*f*−*f*_{0})/*f*_{0}, then as ∂*f*_{0}/∂** c**=−2

*c**f*

_{0}/

*C*

^{2}, the kinetic equation becomesIn a frame moving with the fluid velocity and acceleration, the equation of motion for the fluid (electrons or ions) is already incorporated and in particular the acceleration due to the pressure gradient force is zero in . Hence by (8.2) in steady conditions,Now, expanding

*φ*in a Knudsen number power series gives(8.5)which reduces the problem to finding the solutions of a series of equations for

*φ*

_{1},

*φ*

_{2}, …. The operator

*τD*is

*O*(

*ϵ*) and therefore the leading equation is(8.6)where the collision interval

*τ*

_{2}applies to energy transport and

*τ*

_{1}applies to momentum transport.

The solution of (8.5) is (see Woods 1993)(8.7)whereand is a rather complicated fourth-order tensor, the definition of which is given in Woods (2004, p. 158). Note that the magnetic field has the effect of replacing ∇*T* by .∇*T* and therefore the heat flux vector is(8.8)which should be compared with (7.6).

Unlike the mean free-path treatment, this Boltzmann-like kinetic theory does *not* yield any *O*(*ϵ*^{2}/*ϖ*) terms in tokamak geometry, where the case of interest is *ϖ*≫1 with no parallel gradients. We can therefore assume that =−** b**×/

*ϖ*and a similar reduction applies to . The l.h.s. of (8.6) is

*O*(

*ϵ*) and the operator on the r.h.s. has the effect of a division by

*ϖ*, giving the solution

*φ*

_{1}=

*O*(

*ϵ*/

*ϖ*). Substituting

*φ*=

*φ*

_{1}+

*φ*

_{2}into the kinetic equation and subtracting (8.6) from the result, we getwhere all we need to know about

*A*is its functional dependence. This is similar in form to the equation for

*φ*

_{1}, so its solution will introduce another factor 1/

*ϖ*, i.e.

*φ*

_{2}=

*O*(

*ϵ*

^{2}/

*ϖ*

^{2}) and there is no

*O*(

*ϵ*

^{2}/

*ϖ*) term. Although we have used the BGK model to simplify the analysis, this conclusion also applies to Boltzmann's kinetic theory as there is no essential difference in the mathematical structure.

Boltzmann's collision integral is defined at a point, so that the neighbouring conditions upon which transport depends are found by Taylor series expansions along particle trajectories. This fails in a strong-field magnetoplasma where, in a single collision interval, the trajectories are almost infinitely repeating cycling gyrations. The point that labels the particles in this case does not lie on a particle trajectory and the Taylor series expansion is not relevant.

On discovering the problem with Boltzmann's equation described above, the author looked for a modification of the BGK kinetic equation that would yield the mean free-path result that gave such good agreement with observation. The summary of Woods (1984) reads as follows.A new kinetic equation for the velocity distribution function

*f* is proposed here with the innovation that the term (∂*f*/∂*t*)_{c} representing the effect of particle collisions incorporates time delays extending over the interval between successive 90° collisions. When this theory is applied to magnetoplasmas of the type found in tokamaks, it is found that the time delay has a dramatic effect on the heat flux across the magnetic field, giving results orders of magnitude larger than the classical values and in fair agreement with observations. It appears that non-Markovian effects are of great importance in magnetoplasmas and that they provide answers to some unsolved problems in fusion research.

## 9. Extended mean free-path method

Owing to the *O*(*ϵ*^{2}) failure of BKE, when terms of this or higher order are important, it is necessary to turn back the clock to 1860, i.e. to return to the principle of Maxwell's mean free-path method, but extended to two or more mean free paths in sequence. The principle is explained in Woods (2006, pp. 61–67) for a magnetoplasma; here we shall illustrate the method for a neutral gas.

An important property of ** q** not explicit in Fourier's Law

**=−**

*q**κ*∇

*T*is that owing to the time it takes for the molecules to transport and deposit their energy,

**actually depends on the temperature gradient at time**

*q**τ*

_{2}earlier than the present time

*t*. Thus, Fourier's Law is actually an

*O*(

*ϵ*) approximation to(9.1)

To evaluate the r.h.s. of (9.1), we use the fact that on collisional time scales temperature gradients are *embedded* in the fluid, i.e. they are convected with the fluid and spin with an angular velocity ** Ω**=(1/2)∇×

**. Letting be the usual convective derivative and**

*v***the rate of change of a small embedded vector**

*F***, then**

*F***=**

*F**D*

**−**

*F***×**

*Ω***. If**

*F***′ and**

*v***are the fluid velocities at each end of**

*v***so that**

*F***′=**

*v***+**

*v***.∇**

*F***=**

*v***′+**

*v***.{−×**

*F***}, where is the rate of strain tensor, then correct up to**

*Ω**O*(

*ϵ*

^{2}),

*D*

**≈**

*F***−**

*v***′=**

*v***.−**

*F***×**

*F***, so that**

*Ω***=**

*F***.. Thus, to sufficient accuracy,(9.2)and (9.1) becomes(9.3)**

*F*We expand ** q** as

*q*_{1}+

*q*_{2}, so the second-order heat flux is given by(9.4)where

*τ*

_{2}=1.5

*τ*

_{1}. Comparing the numerical coefficient of .∇

*T*with the corresponding coefficient in Burnett's formula for

*q*_{2}, we find that the Burnett value of 26.2 is reduced to 3.75 by our extended mean free-path (EMFP) calculation. Interpreting this in terms of the number of mean free paths required to allow for the time lag, we find that in Burnett's formula the number is approximately 5, whereas with EMFP it is only approximately 2. It would be remarkable if five mean free paths were really required to transport information about ∇

*T*. The same principle extended to a magnetoplasma yields the formula given in (7.6), which produces results in good agreement with observations of tokamak energy transport.

One of the interesting consequences of the *O*(*ϵ*^{2}) theory of MHD is that in certain circumstances heat flows across the magnetic field *up* the temperature gradient (Woods 2004, pp. 170–175). In Ashbourn & Woods (2006), this mechanism has been successfully applied to the problem of heating the solar corona to temperatures of approximately 2×10^{6} K by energy supplied from the very much cooler chromosphere.

The EMFP theory of transport is semi-empirical in that it does require values for the collision intervals. If the force law acting between colliding molecules is known, then the collision intervals can be safely deduced from Boltzmann's equation, which is required only to first order in *ϵ*.

## 10. Conclusions

The classical theory for momentum and energy transport, based on Boltzmann's equation, has been a great success, but its extension to relatively large values of the Knudsen number is not valid, a limitation not always observed. When taken beyond *O*(*ϵ*), the Chapman–Enskog series expansion introduces spurious terms that are physically unreal, and to obtain relatively accurate *O*(*ϵ*^{2}) terms, it is necessary to adopt a physically transparent method like the EMFP approach. In magnetoplasma dynamics, there is the originally unexpected fact that with strong magnetic fields the *O*(*ϵ*^{2}) terms for cross-field transport are orders of magnitude *larger* than the *O*(*ϵ*) terms, which puts the problem of understanding why tokamaks fail to retain their energy well out of reach of Boltzmann's equation or any variation of it like neoclassical theory.

The kinetic model outlined in this paper may find additional application as the fundamental governing equation in Lattice Boltzmann methods, which, as Zhang *et al*. (2006) point out, currently have difficulties in capturing important gas rarefaction effects (such as Knudsen layers).

## Footnotes

↵† Deceased 15 April 2007.

↵For other forms of the

*E*_{∥}term see Helander & Sigmar (2002, p. 113).↵The equation

*j*_{⊥}=×∇*b**p*/*B*, which follows from the MHD equilibrium condition×*j*=∇*B**p*, is referenced here. The current is said to be*due*to the pressure gradient, i.e. the relation is not taken as being a general stability law due to conservation of momentum, but appears to be given the status of a constitutive relation. However, the terms in this relation have different origins—the current is due to an electric field and the pressure gradient is due to anisotropic particle collisions resulting from temperature and number density gradients.↵There is a persistent misconception that magnetoplasmas are diamagnetic; in fact they are very weakly paramagnetic (see Bleaney & Bleaney (1985, p. 473), where it is shown that free electrons have a positive susceptibility of ∼10

^{−8}). Were a plasma really magnetic,would not equal*H**μ*_{0}and the resulting magnetic polarization would add a current*B**μ*_{0}∇to Ampere's law ∇×*×M*=*B**μ*_{0}, but on p. 33 of their text Hazeltine & Meiss write*j**μ*=*μ*_{0}, which means that they should know that=0.*M*↵In fact with

*τ*given by the usual formula, this second-order term is much too large to explain the observations and it became clear from physical considerations that*τ*should be replaced by the much smaller ‘bounce time’ of particles trapped in banana orbits. Later it was found that the collision interval*τ*is appropriate for the sudden avalanche of energy that occurs during a major disruption.- Received July 12, 2006.
- Accepted March 12, 2008.

- © 2008 The Royal Society